A sphere is inscribed in a cube. What is the ratio of the volume of the inscribed sphere to the volume of the cube? Express your answer as a common fraction in terms of $\pi$.
Solution: [asy]
size(60);
draw(Circle((6,6),4.5));
draw((10.5,6)..(6,6.9)..(1.5,6),linetype("2 4"));
draw((10.5,6)..(6,5.1)..(1.5,6));
draw((0,0)--(9,0)--(9,9)--(0,9)--cycle);
draw((0,9)--(3,12)--(12,12)--(9,9));
draw((12,12)--(12,3)--(9,0));
draw((0,0)--(3,3)--(12,3),dashed); draw((3,3)--(3,12),dashed);
[/asy]

Let the side length of the cube be $s$.  The side length of the cube is equal to diameter of the inscribed sphere, so the radius of the sphere has length $\frac{s}{2}$.  Thus, the volume of the sphere is equal to $\frac{4}{3}\pi \left(\frac{s}{2}\right)^3 = \frac{\pi s^3}{6}$ and the volume of the cube is equal to $s^3$.  Hence the ratio of the sphere's volume to the cube's volume is $\boxed{\frac{\pi}{6}}$.